Recipes and Economic Growth: A Combinatorial March Down an Exponential Tail Chad Jones Stanford GSB December 3, 2020 0
Combinatorics and Pareto • Weitzman (1998) and Romer (1993) suggest combinatorics important for growth. ◦ Ideas are combinations of ingredients ◦ The number of possible combinations from a child’s chemistry set exceeds the number of atoms in the universe ◦ But absent from state-of-the-art growth models? • Kortum (1997) and Gabaix (1999) on Pareto distributions ◦ Kortum: Draw productivities from a distribution ⇒ Pareto tail is essential ◦ Gabaix: Pareto distribution (cities, firms, income) derived from exponential growth Chicken and egg problem: Which comes first, Pareto distn or exponential growth? 1
Two Contributions • A simple but useful theorem about extreme values ◦ The increase of the max extreme value depends on (1) the way the number of draws rises, and (2) the shape of the upper tail ◦ Applies to any continuous distribution • Combinatorics and growth theory ◦ Combinatorial growth: Cookbook of 2 N recipes from N ingredients, with N growing exponentially (population growth) Combinatorial growth with draws from thin-tailed distributions (e.g. the normal distribution) yields exponential growth ◦ Pareto distributions are not required — draw faster from a thinner tail 2
Basic Foundations 3
Theorem 1 (A Simple Extreme Value Result) Let Z K denote the max over K i.i.d. draws from a continuous distribution F ( x ) , with ¯ F ( x ) ≡ 1 − F ( x ) strictly decreasing. Then K ¯ � � = e − m K →∞ Pr lim F ( Z K ) ≥ m As K increases, the max Z K rises precisely to stabilize K ¯ F ( Z K ) . The shape of the tail of ¯ F ( · ) and the way K increases determines the rise in Z K 4
Graphically ¯ F ( x ) ¯ F ( Z K ) = Pr [ Next draw > Z K ] ¯ F ( Z K ) ¯ F ( x ) x 0 0 Z K 5
Graphically ¯ F ( x ) ¯ F ( Z K ) = Pr [ Next draw > Z K ] More draws K means Z K increases and ¯ F ( Z K ) declines. – Good ideas get harder to find! ¯ F ( Z K ) ¯ F ( Z ′ K ) ¯ F ( x ) x 0 0 Z K Z ′ K 5
Graphically ¯ F ( x ) ¯ F ( Z K ) = Pr [ Next draw > Z K ] More draws K means Z K increases and ¯ F ( Z K ) declines. – Good ideas get harder to find! Blowing up by K ⇒ K ¯ F ( Z K ) will stabilize. So the rate at which K increases and the tail of ¯ F ( · ) determine how Z K rises... ¯ F ( Z K ) ¯ F ( Z ′ K ) ¯ F ( x ) x 0 0 Z K Z ′ K 5
Proof of Theorem 1 • Given that Z K is the max over K i.i.d. draws, we have Pr [ Z K ≤ x ] = Pr [ z 1 ≤ x , z 2 ≤ x , . . . , z K ≤ x ] = ( 1 − ¯ F ( x )) K • Let M K ≡ K ¯ F ( Z K ) denote a new random variable. Then K ¯ � � Pr [ M K ≥ m ] = Pr F ( Z K ) ≥ m F ( Z K ) ≥ m � � ¯ = Pr K F − 1 � m � � � Z K ≤ ¯ = Pr K 1 − m � K � → e − m = QED. K 6
Remarks • Simpler and different from the standard EVT ◦ If Z K − b K converges in distribution, then it converges to one of three types a K ◦ Which one depends on the tail properties of F ( · ) • We will see later that Theorem 1 covers cases not covered by EVT • Intuition for why so few conditions on F ( · ) are required: ◦ For any distribution of x , ¯ F ( x ) is Uniform[0,1] ◦ Min over K draws from a uniform, scaled up by K , is exponential = K ¯ F ( Z K ) (from standard EVT) ◦ Galambos (1978, Chapter 4) has related results 7
Example: Kortum (1997) • Pareto: ¯ F ( x ) = x − β • Apply Theorem 1: K ¯ F ( Z K ) = ε + o p ( 1 ) KZ − β = ε + o p ( 1 ) K K = ε + o p ( 1 ) Z β K Z K K 1 /β = ( ε + o p ( 1 )) − 1 /β • Exponential growth in K leads to exponential growth in Z K g Z = g K /β β = how thin is the tail = rate at which ideas become harder to find 8
Example: Drawing from an Exponential Distribution • Exponential: ¯ F ( x ) = e − θ x K ¯ F ( Z K ) = ε + o p ( 1 ) Ke − θ Z K = ε + o p ( 1 ) log K − θ Z K = log( ε + o p ( 1 )) ⇒ Z K = 1 θ [log K − log( ε + o p ( 1 ))] ⇒ Z K � 1 − log( ε + o p ( 1 )) � log K = 1 ⇒ θ log K Z K p → Constant − log K 9
Drawing from an Exponential (continued) Z K p → Constant − log K • Z K grows with log K ◦ If K grows exponentially, then Z K grows linearly • Definition of combinatorial growth: K t = 2 N t with N t = N 0 e g N t g Z = g log K = g N Combinatorial growth with draws from a thin-tailed distribution delivers exponential growth! 10
Growth Model 11
Setup • Cookbook is a collection of K t recipes • At a point in time, researchers have evaluated all recipes from N t ingredients ◦ Each ingredient can either be included or excluded, so K t = 2 N t (which equals � N t � N t � , the sum of all combinations) k = 0 k • Research = learning the “productivity” of the new recipes that come from adding a new ingredient • Note: if ∆ N t + 1 = α R t , then each researcher can evaluate α new ingredients each period ◦ R t grows with population ⇒ so does N t Combinatorial growth: Cookbook of K = 2 N recipes from N ingredients, with N growing exponentially 12
Economic Environment σ �� 1 σ − 1 � σ − 1 Y t = 0 Y σ di with σ > 1 Aggregate output it ρ � � − 1 ρ − 1 ρ − 1 � M it ρ ρ Y it = Z Kit M j = 1 x di with ρ > 1 Variety i output it ijt x ijt = L ijt Production of ingredients Z Kit = max c z ic , c = 1 , ..., K t Best recipe z ic ∼ F ( x ) = 1 − e − x β Weibull distribution of z ic α R λ like ˙ N t = α R λ t N φ ∆ log N t + 1 = t Number of ingredients evaluated N 1 − φ t t K t = 2 N t Cookbook � 1 L it = � M i j = 1 L ijt and 0 L it di = L yt Resource constraint: workers R t + L yt = L t Resource constraint: R&D L t = L 0 e g L t Population growth (exogenous) 13
Allocation • Consider the allocation of labor that maximizes Y t at each date with a constant fraction of people working in research ◦ L ijt maximizes Y t ◦ R t = ¯ sL t split symmetrically • Number of ingredients evaluated (eventually) grows at a constant rate g N = λ g L 1 − φ • And we have combinatorial growth in the number of recipes in the cookbook K t = 2 N t ⇒ g log K = g N 14
Applying Theorem 1 to the Weibull Distribution • Suppose y ∼ Exponential. Let y ≡ x β . Then x ∼ Weibull: ¯ F ( x ) = e − x β max y p → Constant − log K max x β p → Constant ⇒ − log K max x p → Constant ⇒ − (log K ) 1 /β • Therefore g Z K = g log K = g N λ g L β = 1 β β 1 − φ 15
Remarks g Z K = g log K = g N λ g L β = 1 β β 1 − φ • This is the growth rate of output per person in the growth model • Combinatorial march down a Weibull tail • Growth rate depends on ◦ Population growth = growth rate of researchers ◦ λ and φ = how researchers evaluate ingredients ◦ Allows φ > 0 : it may get easier (or harder) to evaluate ingredients ◦ While β captures the degree to which good ideas get harder to find 16
Can the distribution shift out over time? • Consider all the technologies that could ever be invented. They are recipes. ◦ Let ¯ F ( x ) be the associated distribution of productivities ◦ That doesn’t shift... • What’s behind the question: some technologies cannot be invented before others ◦ The smartphone could not come before electricity, radio, and semiconductors • Easy to incorporate: suppose the ingredients must be evaluated in a specific order ◦ Nothing changes... ◦ (Note: evaluation can get easier or harder over time, via ∆ N t + 1 = α R t N φ t ) 17
Generality? For what distributions do combinatorial draws ⇒ exponential growth? 18
Theorem 2 (Necessary and sufficient conditions) Consider the setup in Theorem 1, and suppose K exhibits combinatorial growth, i.e. K t = 2 N t and N t = N 0 e g N t . F ( x ) ; that is, η ( x ) ≡ − d log ¯ Let η ( x ) denote the elasticity of the tail cdf ¯ F ( x ) d log x . Then → g N p ∆ log Z Kt − α if and only if η ( x ) lim = Constant > 0 x α x →∞ for some α > 0 . 19
Remarks → g N η ( x ) p ∆ log Z Kt lim = Constant > 0 − ⇐ ⇒ x α α x →∞ • Kortum (1997): ¯ F ( x ) = x − β ⇒ η ( x ) = β so exponential growth in K is enough • Thinner tails require faster draws but still require power functions: ◦ It’s just that the elasticity itself is now a power function! • Examples F ( x ) = e − x β ⇒ η ( x ) = x β ◦ Weibull: ¯ � x −∞ e − u 2 / 2 du ⇒ η ( x ) ∼ x 2 – like Weibull with β = 2 ◦ Normal: ¯ F ( x ) = 1 − 20
For what distributions do combinatorial draws ⇒ exponential growth? • Combinatorial draws lead to exponential growth for many familiar distributions: ◦ Normal, Exponential, Weibull, Gumbel ◦ Gamma, Logistic, Benktander Type I and Type II F ( x ) = x α e − x β or ¯ ◦ Generalized Weibull: ¯ F ( x ) = e − ( x β + x α ) ◦ Tail is dominated by “exponential of a power function” • When does it not work? ◦ lognormal: If it works for normal, then log x ∼ Normal means percentage increments are normal, so tail will be too thick! ◦ logexponential = Pareto ◦ Surprise: Does not work for all distributions in the Gumbel domain of attraction (not parallel to Kortum/Frechet). 21
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